Maximum value a function satisfying a differential equation can achieve.

[caffeinemachine]

Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line?

a) 4
b) 3
c) 5
d) Maximum value does not exist.

I am quite lost on this one. After some thought I am convinced that the maximum value should exist, though I do not have a good argument to support this claim.

 

[I like Serena]

When I throw it at Octave online, I get:

View attachment 9668

So it seems it is none of the above, but the answer $3$ is close.

For reference, I have re-encoded the differential equation a bit to solve it with Octave's [M]lsode[/M].

Let $y=f(x)$, $u=y'$, and $Y=(u,y)$ then:

$f(x)+f''(x)=-x|\sin x|f'(x) \\
y + u' = -x|\sin x|u \\
\begin{cases}u'=-y-x|\sin x|u\\ y'=u\end{cases}\\
(u', y') = (-y-x|\sin x|u,\, u) =: g(u,y;x)\\
Y' = g(Y;x)
$